8 Topology Questions : Hausdorff Space, Countability, Compactness and Homomorphisms

1. Show that the collection
13 {[O,c)J 0<c 1}u{(d,1]|O<d< 1)
cannot be a base for the subspace topology [O, 1], where is the Euclidean topology on rt
Hint: Use contradiction, i.e., first assume that B is a base for [O, 1].
2. Let B {[a, b) x [c, d)Ia < b, : < d}. Show that B is a base for the product space (R x R, £ x £), where £ is the lower limit topology on 1?.
3. Let X. Y be spaces and f : X ?> Y a continuous surjcction. Prove that if X is P countable, so is Y.
4. Let X, Y he spaces and f : X ?> Y a continuous surjection. Prove that if X is compact, so is V.
3. Show that the union of finitely many compact subsets of a space is compact.
6. Let X be a compact space and V a Hausdorff space. Then show that f X ?> Y is a homomorphism if and only if f is bijeetive and continuous.
7. Let X be a Hausdorff space and P, Q disjoint compact subsets of X. Then show that there exist open sets U and V such that P C U, Q C V and U U V 0.
This property tells us that compact subsets of a Hausdorif space behaves like points.
8. Let X, Y be spaces arid f X ?> V a continuous surjection. Prove that if X is connected, so is V.

2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

Let X and Y connected, locally path connected and Hausdorff. let X be compact.
Let f: X ---> Y be a local homeomorphism. Prove that f is a surjective covering with finite
fibers.
Prove:
a) Any subspace of a weak Hausdorff space is weak Hausdorff.
b)Any open subset U of a compactly generated space X is compactly generated

The question we want to answer is as follows. For a nonempty compact Hausdorff topological space X and a continuous function f:X-->X we want to show that there is a fixed set A for f, that is, A is nonempty and f(A)=A.
We also construct an example of a Hausdorff space X which is not compact for which there are no fixed sets,

I have these problems from Topology of Surfaces by L.Christine Kinsey: the problems I require assistance with are 2.26, 2.28, 2.29, and 2.32. These are stated below.
PROBLEM (Exercise 2.26). Describe what stereographic projection does to
(1) the equator,
(2) a longitudinal line through the north and south poles,
(3) a tr

19. Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology of Y.
(question is also included in attachment)

Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y. Prove that J indeed a toplogy on Y i.e., (Y,J) is a topological space.

Please see the attached file for the fully formatted problems.
B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant.
(b) State what it means for a map f X -?> Y to be open. Show that a continuous open bijection is a homomorphism.
(c) (i) Recall that Fr E, the fron

Let R be a Boolean ring and let X be the set all maximal ideals of R. Put a topology on X by taking sets of the form Dr = {M ∈ X | r ∈/ M}, r is in R, as a basis for the open topology. Since the ideals are prime, Dr ∩ Ds = Drs, making the collection closed under finite intersections. Show that X is a Boolean space. Hint on